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The function which takes the value 0 for rational number and 1 for irrational number (cf. Dirichlet function) is bounded. Thus, a function does not need to be "nice" in order to be bounded. The set of all bounded functions defined on [ 0 , 1 ] {\displaystyle [0,1]} is much larger than the set of continuous functions on that interval.
A set with an upper (respectively, lower) bound is said to be bounded from above or majorized [1] (respectively bounded from below or minorized) by that bound. The terms bounded above ( bounded below ) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.
The metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness. For subsets of R n the two are equivalent. A metric space is compact if and only if it is complete and totally bounded. A subset of Euclidean space R n is compact if and only if it is closed and
A sequence = = is said to be Mackey convergent to the origin in if there exists a divergent sequence = = of positive real number such that = = is a bounded subset of . if X {\displaystyle X} and Y {\displaystyle Y} are locally convex then the following may be add to this list:
Briefly, a closed set contains all of its boundary points, while a set is bounded if there exists a real number such that the distance between any two points of the set is less than that number. In R {\displaystyle \mathbb {R} } , sets that are closed and bounded, and therefore compact, include the empty set, any finite number of points, closed ...
Bounded set (topological vector space), a set in which every neighborhood of the zero vector can be inflated to include the set; Bounded variation, a real-valued function whose total variation is bounded; Bounded pointer, a pointer that is augmented with additional information that enable the storage bounds within which it may point to be deduced
Each set has a supremum (infimum), if it is bounded from above (below). Proof: Without loss of generality one can look at a set A ⊂ R {\displaystyle A\subset \mathbb {R} } that has an upper bound. One can now construct a sequence ( I n ) n ∈ N {\displaystyle (I_{n})_{n\in \mathbb {N} }} of nested intervals I n = [ a n , b n ] {\displaystyle ...
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family.